Spherical aberration (3 rd order)
Aberration function
( ) 4
040
222
040 ρWyxWW PP =+=
•Independant of the field (y’) : we study it on the axis
•Independant of ϕ : we study it in the tangential plane (0z’y’)
•HP : the pupil is on the component (diopter, lens, mirror)
•We’ll see later that the spherical aberration does not depend on the position of the pupil
2Thierry Lépine - Optical design
X
Y
Z
Description
Paraxial image
Paraxial focus
Marginal focus
Axial caustic
Tangential caustic
Longitudinal spherical
aberration (LA)
Marginal ray
Least confusion
circle
Paraxial ray
3Thierry Lépine - Optical design
Formalism
( ) 4
040
222
040 ρWyxWW PP =+=( )
( ) 3
040)(
22
040
222
040
44 ρ
ε
WRn
RyyxW
Rn
R
y
yxW
Rn
R
y
W
Rn
R
PyzO
PPP
P
P
PP
PPP
y
××′
−=+××′
−=
∂+∂×
×′−=
∂∂×
×′−=
′′
A’0
A’’
I
Σ’0Σ’
JxP
yP
z’
Paraxial image
Exit pupil
LA
εy
5Thierry Lépine - Optical design040
1
3
040
8
8 : image theofdiameter theHence WRn
RW
Rn
R
PP′
=′
=Φ=ρ
ρ
Formalism
( ) 040
2
1
2
040
2
2
040
2
22
2
4
040)(
14 : Hence
14
140
: Hence
2
n
:naught is aberration e transversthefor which defocus theis
what know toneed we(LA), aberration allongitudin theevaluate To
WR
R
nLA
WR
R
nyW
R
R
ny
W
Rn
R
yRR
yWW
P
z
P
P
P
z
PP
y
PPz
PyzO
′−==
′−=
′−=⇒=
∂∂
′−=
××
′+=
=
′′
ρε
ρεε
ε
A’0
A’’
I
Σ’0Σ’
JxP
yP
z
Paraxial image
Exit pupil
LA=εz
εy
6Thierry Lépine - Optical design
Aberration function(the reference sphere is centered on the paraxial image)
λ3040 =W
7Thierry Lépine - Optical design
Viewing the PSF (star test)
Thierry Lépine - Optical design 8
λ3040 =W
-40
-35
-30
-25
-20
-15
-10
-5
0
-40
-35
-30
-25
-20
-15
-10
-5
0
MTF
Thierry Lépine - Optical design 11
λ3040 =W
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f (cyc/mm)
MT
F
MTF(u)
MTF(v)
diffraction
Transverse ray aberrations (paraxial image plane)
12Thierry Lépine - Optical design
-1 0 10
0.2
0.4
0.6
0.8
1
yp
Wy
y'=1
0 0.5 10
0.2
0.4
0.6
0.8
1
xp
Wx
y'=1
-1 0 1-10
-5
0
5
10
yp
epsy
y'=1
0 0.5 1-8
-6
-4
-2
0
xp
epsx
y'=1
-1 0 10
0.2
0.4
0.6
0.8
1
yp
Wy
y'=0.7
0 0.5 10
0.2
0.4
0.6
0.8
1
xp
Wx
y'=0.7
-1 0 1-10
-5
0
5
10
yp
epsy
y'=0.7
0 0.5 1-8
-6
-4
-2
0
xp
epsx
y'=0.7
-1 0 10
0.2
0.4
0.6
0.8
1
yp
Wy
y'=0
0 0.5 10
0.2
0.4
0.6
0.8
1
xp
Wx
y'=0
-1 0 1-10
-5
0
5
10
yp
epsy
y'=0
0 0.5 1-8
-6
-4
-2
0
xp
epsx
y'=0
Circle of least confusion
( )
2
3
2
1
040
2
2
3
3
040
2
0402
2
3
0401n
3
1 : caustic theofequation cartesian then theand
8et 12
:branch)(upper caustic theof equations parametric thededucecan weSo
0 :point contact for the hence caustic, theo tangent tisray This
4y
: b and a constants theof sexpression thededucecan weSo
0et
0 points he through tgoesray This
: pupil (exit) thefrom emergingray afor Equation
zWR
Ry
WR
RyW
R
Rz
dρ
yd
zR
RW
R
RzLA
LA
LA
bazy
P
PP
P
P
−
=′
−=′
=′−=
=′
−−=−=′
+=′
ρρ
ρρε
ε
ρρ
ρ
ρ
ρ
13Thierry Lépine - Optical design
2 1 3
2 2040 0403(lower branch)
2
rd
At the circle of least confusion (CLC), the marginal ray ( 1) intersects the negative part
of the caustic :
14
3
This is a 3 degree equation of unk
P P
P
R RRW z W z
R R R
ρ
−
=
− − = − −
1
nown z.
3We just check that LA is a solution.
4
The image position for the circle of least confusion occurs three-quarters of the way from
the paraxial focus to the marginal focus.
The diameter of th
z ρ ==
1
1
3 040LA
4
24 2 4
040 0402 3LA
4
e CLC is : 2 2 . It is the quarter of the diameter
of the paraxial image.
If the center of the reference sphere is at the center of the CLC :
1,52
z
zP
Pz
Ry W
R
RW W W
R
ρ
ρερ ε ρ ρ ρ
=
=
=
=
′ =
= + = −( )2
Circle of least confusion
14Thierry Lépine - Optical design
Best focus
! foci marginal and paraxial ebetween thhalfway is focusbest The
! 2
2
2 :or , : Hence
,0 ie. minimum, is for which (b) defocus for the looking are We
12645
4
325
1W
23
1
W
2 then defocus, a introduce weIf
minimum. is ie. maximum, is ratio Strehl thefocus,best At the
0402
2
0402
2
22
222
222
0
1
0
22
2
0
1
0
222
242
2
24
0401
2
LAW
R
RW
R
Rab
b
baba
babaddW
baddWW
W
baR
RWW
P
zP
z
WW
W
W
Pz
n
W
=−=⇒−=−=
=∂
∂
++=⇒
++==
+==
−=
+=+=
∫ ∫
∫ ∫
=′
εε
σσ
σ
ϕρρπ
ϕρρπ
σ
ρρρερ
σ
π
π
15Thierry Lépine - Optical design
Best focus
( )( )
( )
pupil. theof edges at the and
(obvious) axis on thenaught is focusbest for thefunction aberration that thenote alsocan We
sphere. reference therespect to with retarded, is wavefront the0, Wif and,
image, paraxial at the valueitsan smaller th times4 is WFEmaximum thefocus,best At the
42
10 :maximum is for which for looking are We
: casesboth for functions aberration theof valuemaximum thedetermine usLet
040
040
max
24
040
0401max
4
040
>
=⇒=⇒=∂
∂−=
=⇒
=
=
WW
WW
WW
WW
WW
BFBF
BF
BF
paraxial
paraxial
ρρ
ρ
ρρ
ρ
ρ
16Thierry Lépine - Optical design-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
y
W paraxial
W best focus
Aberration function(reference sphere centered at best focus)
λλ 3,3 020040 −== WW
17Thierry Lépine - Optical design